Basic Description

The law of cosines is a formula that helps in solving triangles when two or three side lengths of a triangle are known.

Given a triangle with side lengths, and angle measures , the law of cosines states

The formula combines the squares of two side lengths of a triangle and and a third term involving the cosine of a particular angle, to calculate the square of the third side. For this reason, the law of cosines is often thought of as the generalization of the Pythagorean theorem, which only applies to right triangles. The law of cosines adds an extra term to the Pythagorean theorem so that a third side length of a triangle can be determined when there is no right angle.

The law of cosines is useful in solving triangles whenever at least two side lengths are known. If the triangle that needs to be solved has less than just two known lengths, the law of sines is more useful.

Alternate Forms

When the law of cosines is being used to find a particular side length, that is when a certain angle measure is given, the law of cosines can be written in several different ways to help set up the equation to solve for the missing element of the triangle.

An alternate form of the law of cosines is particularly useful when solving a triangle when just the three side lengths are given. This form isolates the term with the cosine of the angle in it to make it easier to solve for the angle.

A More Mathematical Explanation

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Proof

By the Pythagorean Theorem

One way to think of the law of cosines is as an extension [...]

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Proof

By the Pythagorean Theorem

One way to think of the law of cosines is as an extension of the Pythagorean theorem for a right triangle:

By Pythagorean theorem, we know

But is just some portion of side length which is less than the length of . Substituting the difference gives us,

By Pythagorean theorem, we also know that

Substituting the appropriate values gives us,

Expanding the squared term gives us

Simplify for

And by the definition of cosine, we know that

Substituting this value in give us

or

Using the Distance Formula

The law of cosines solves for a particular side length using the other side lengths and an angle. We can write this length using the distance formula as the distance from one vertex of the triangle to another.

Let be oriented so that is at the origin, and is at the point.

We use the formula for the distance between two points

is the distance from to .

Since and , substituting the appropriate points into the distance formula gives us

Squaring the inner terms, we have

Since ,

Square both sides for

Example Problem

Solve the triangle using the law of cosines.

Solution

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To find the side length ,

Simplify for

Since , substitution gives us

Simplify for

Taking the square root of both sides gives us

To solve for angle , we can orient the triangle differently and use the law of cosines in the form